Some More Updates on an Annihilation Number Conjecture: Pros and Cons

Abstract

If $\alpha (G)+\mu (G)=\left| V\right|$, then $G=\left( V,E\right)$ is a König–Egerváry graph, where $\alpha (G)$ denotes the cardinality of a maximum independent set, while $\mu (G)$ is the size of a maximum matching in G. If $d_{1}\le d_{2}\le \cdots \le d_{n}$ is the degree sequence of G, then the annihilation number $a\left( G\right)$ of G is the largest integer k such that $\sum \limits _{i=1} ^{k}d_{i}\le \left| E\right|$ (Pepper, Binding independence, Ph.D. Dissertation, University of Houston, 2004; Pepper, On the annihilation number of a graph, in: Recent Advances in Electrical Engineering: Proceedings of the 15th American Conference on Applied Mathematics, pp 217–220, 2009). A set $A\subseteq V$ satisfying $\sum \limits _{a\in A}deg (a)\le \left| E\right|$ is an annihilation set; if, in addition, $deg \left( v\right) +\sum \limits _{a\in A}deg (a)>\left| E\right|$, for every vertex $v\in V(G)-A$, then A is a maximal annihilation set in G. In Larson and Pepper (Graphs with equal independence and annihilation numbers. Electron J Comb 18:180, 2011) it was conjectured that the following assertions are equivalent: (i) $\alpha \left( G\right) =a\left( G\right)$; (ii) G is a König–Egerváry graph and every maximum independent set is a maximal annihilating set. Recently, it turned out that the implication “(i) $\Longrightarrow$ (ii)” was not true. A series of corresponding counterexamples can be found in Hiller (Counterexamples to the characterisation of graphs with equal independence and annihilation number. arXiv:2202.07529v1 [math.CO], 2022). In Levit and Mandrescu (On an annihilation number conjecture. Ars Math. Contemp. 18, 359–369, 2020), we presented an infinite family of non-bipartite König–Egerváry graphs that invalidate the “ (ii) $\Longrightarrow$ (i)” part of this conjecture. In this paper, we provide two more infinite families of counterexamples, one consisting of trees and the other one comprising non-tree bipartite graphs. We also show that the above conjecture is true for trees with $\alpha \left( G\right) =4$, disconnected non-bipartite König–Egerváry graphs with $\alpha \left( G\right) =4$, and disconnected bipartite graphs with $\alpha \left( G\right) =4$ excluding the three following counterexamples: $C_{4}\cup 2K_{2},Domino\cup K_{2}$ and $K_{3,3}-e$.